sol = Solve[{l1^2 == x1^2 + x2^2, l2^2 == y1^2 + y2^2, p1 == x1 + y1,
p2 == x2 + y2}, {x1, y1, x2, y2}] //
FullSimplify[#, p2 > 0 && p1 > 0] &;
Manipulate[
Graphics[
Table[With[{mp1 = 1.3 ml1 - 2*Cos[T]*(Cos[T] + α)/4 + 1/4,
mp2 = Sin[T]*(Sin[T] + β)/4 + 1/4},
{RGBColor[0, 0, 0, ((T - t)/10 π)^2],
Line[{{0, 0}, {x2, -x1}, {x2 + y2, -x1 - y1}}]} /. # & /@
sol[[{1}]] /. {p1 -> mp1, p2 -> mp2, l1 -> ml1, l2 -> ml2}
], {T, t, t + 2 π, 2 π/10}],
PlotRange -> {{-10, 10}, {-10, 0}}
],
{ml1, 6, 10},
{ml2, 3, 10},
{t, 0, 2 π, Appearance -> "Open"},
{{α, 2}, 2, 4},
{{β, 10}, 2, 10}
]